管状区域中非线性扩散问题的行波行为
Travelling wave behaviour arising in nonlinear diffusion problems posed in tubular domains
摘要：
对于固定的有界区域$D \subset \mathbb{R}^N$，我们研究了管状区域\begin{equation*} \partial_t u = Δ_p u \quad \text{ in } D \times \mathbb{R}, \quad t > 0 \end{equation*}中$p$ -Laplacian扩散方程解的大次数渐近行为，即，慢扩散的情况下，和齐次Dirichlet边界条件的管边界。传递到合适的重新缩放的变量，我们显示了在对数时间的行波解的存在，连接水平$u = 0$和唯一的非负稳态相关的重新缩放的问题，在较低的维度，即在$D\subset \mathbb{R}^N$。然后，我们采用这种特殊的波表明，广泛的一类解决方案收敛到通用的固定配置文件中的管的中间，并在同一时间，他们在两个轴向管的方向传播，模仿行为的行波（及其反射）大的时间。我们的分析的第一个主要特点是，波前是通过一个（非标准的）组合的扩散和吸收边界条件，这引起了一种费舍尔KPP长时间的行为。第二，非线性扩散项在我们的分析中起着至关重要的作用。实际上，在线性扩散框架中，$p=2$解决方案的表现完全不同。
英文摘要：
For a fixed bounded domain $D \subset \mathbb{R}^N$ we investigate the asymptotic behaviour for large times of solutions to the $p$-Laplacian diffusion equation posed in a tubular domain \begin{equation*} \partial_t u = Δ_p u \quad \text{ in } D \times \mathbb{R}, \quad t > 0 \end{equation*} with $p>2$, i.e., the slow diffusion case, and homogeneous Dirichlet boundary conditions on the tube boundary. Passing to suitable re-scaled variables, we show the existence of a travelling wave solution in logarithmic time that connects the level $u = 0$ and the unique nonnegative steady state associated to the re-scaled problem posed in a lower dimension, i.e. in $D\subset \mathbb{R}^N$. We then employ this special wave to show that a wide class of solutions converge to the universal stationary profile in the middle of the tube and at the same time they spread in both axial tube directions, miming the behaviour of the travelling wave (and its reflection) for large times. The first main feature of our analysis is that wave fronts are constructed through a (nonstandard) combination of diffusion and absorbing boundary conditions, which gives rise to a sort of Fisher-KPP long-time behaviour. The second one is that the nonlinear diffusion term plays a crucial role in our analysis. Actually, in the linear diffusion framework $p=2$ solutions behave quite differently.
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